Problem 8 - College Algebra and Calculus: An Applied Approach

Question

Determine whether the system of equations is in row-echelon form. Justify your answer. $$ \left\\{\begin{array}{rr} x-y-8 z= & 12 \\ 2 y-2 z= & 2 \\ 7 z= & -7 \end{array}\right. $$

Step-by-Step Solution

Answer

Yes, the given system of equations is in row-echelon form.

Step 1: Understand row echelon form

A system of linear equations is in row-echelon form if all non-zero rows come before the rows that are all zeros (if any), and the leading coefficient (the first non-zero number from the left, also called the pivot) of a non-zero row is always to the right of the leading coefficient of the row above it.

Step 2: Compare the given system of equations with row-echelon form

In the given system of equations:\[\begin{align*}x - y - 8z &= 12, \2y - 2z &= 2, \7z &= -7\end{align*}\]The first non-zero number in each equation (the leading coefficient) moves one position to the right, and no equation consists only of zero. Therefore, the system of equations is in row-echelon form.

Key Concepts

Linear Equations

Linear equations are fundamental concepts in mathematics that allow us to represent relationships between different quantities. They are equations of the first order, meaning each term is either a constant or the product of a constant and a single variable. Typically, they appear in forms like:

• Single equations or a series of equations.
• Each equation represents a straight line when graphed in a two-dimensional plane.

In the context of systems of linear equations, like the one provided, the goal is often to find the values for the variables that satisfy all equations simultaneously. These systems can be solved using various methods, such as substitution, elimination, or matrix operations, which is where row-echelon form becomes crucial.
Scientists and mathematicians use linear equations to model problems in physics, economics, and engineering, simplifying complex real-world phenomena into manageable solutions.

Pivot Element

In linear algebra, a pivot element plays a vital role in transforming a matrix to row-echelon or reduced row-echelon form. The pivot element is the first non-zero number in each row of a matrix, and it is also referred to as the leading term.
When working with systems of linear equations, identifying the pivot element helps determine whether a system is in row-echelon form:

• The pivot must appear to the right of any pivot in a row above it.
• Each pivot helps in zeroing out the elements below it, simplifying the system.

This step-by-step reduction is essential for methods like the Gaussian elimination, where the calculation focuses on moving pivot elements and clearing columns beneath them. Locating and managing pivot elements facilitate the simple progression from complex systems of equations to easily interpretable solutions.

Leading Coefficient

The leading coefficient in a system of linear equations is another term for the pivot element in a matrix row. It is the first non-zero number in a row when reading from left to right. Understanding this concept is key to determining the structure of the row-echelon form.
This leading coefficient defines each row's role in a linear system:

• In a row-echelon form, the leading coefficients descend in a stair-step pattern from one row to the next.
• Shifting the leading coefficient advances the transformation towards solving the equations, crucial for precise calculations.

The position and value of leading coefficients affect the system's nature and the solutions possible, helping cement a foundation for further mathematical exploration in linear algebra.